PYTS/ASTR 206 – Orbits and Gravity 1
Announcements D2L site up
Homework 1 posted after class on website http://www.lpl.arizona.edu/~shane/PTYS_206
Look at the homework early! You have a week to finish – not a week to start! Due in class next Thursday
TA is only Kevin Jones for now Priyanka will join us in ~3 weeks – please don’t visit her until then. Office hours: Kevin – Tuesday 2-4pm, Gould-Simpson 511
Myself – Tue./Thur. 1.45-4pm, Kuiper 524
PYTS/ASTR 206 – Orbits and Gravity 2
Observations Craters, Trenches, Bright/white surface Higher ground on the right, more craters on the left
PTYS/ASTR 206 – The Golden Age of Planetary Exploration
Shane Byrne – [email protected]
Orbits and Gravity
PYTS/ASTR 206 – Orbits and Gravity 4
Gravity Newton and Galileo
Planetary Shape Flattening The Geoid
Tides Fate of the Moon
Orbits of planetary objects Kepler’s laws
In this lecture…In this lecture…
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Gravity is one of 4 physical forces in nature
Relatively unimportant on small scales …but it dominates the universe at large
scales
Where does gravity operate?Where does gravity operate?
Holds galaxies together
Provides pressure for fusion reactions in stars
Holds planets in their orbits
Holds planets together
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Objects that contain matter (they have mass) attract each other This attractive force is called gravity, e.g.
Your body and the Earth The Moon and the Earth The Earth and the Sun Your body with a comet billions of km away
The force of this attraction depends on three things Mass of the first object
Mass of the second object
How far apart the objects are – e.g. distance between center of planet and objects on the surface
On Earth this force is the weight of the first object
What’s gravity?What’s gravity?
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Galileo was first to systematically investigate gravity
Objects accelerate as they fall i.e. the longer the drop the faster they hit the ground
All objects accelerate at the same rate The famous leaning tower of Pisa experiment Some doubt about whether this actually happened
This acceleration is 9.8 m/s2
0 m/s when you let it go 9.8 m/s after 1 seconds 2*9.8 m/s after 2 seconds etc…
This acceleration is not the same on other planets
e.g. it’s 3.7 m/s2 on Mars
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F = M1 * a1
Fgravity = M1 * agravity
What gives? Is gravity a force or an acceleration? It’s a force that produces an acceleration
Isaac Newton developed modern mechanics – using three basic laws
With no forces operating an object doesn’t change its momentum (m1 * v1 )
A force (F) causes a change in momentum F = M1 * v1 – M1 * v2 = M1 * (v1 – v2) = M1 * a1
When one body exerts a force on another then there is an equal an opposite reaction
My Fgravity = 80 kg * 9.8m/s2
784 Newtons (or ~175 lbs)
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Newton also figured out how to calculate the gravitational force between objects.
Mass of object 1 (M1) Mass of object 2 (M2) Distance between them (R)
Objects behave like points at their centers of gravity
M1 M2
R
Fgravity = G * M1 * M2
R2
‘G’ is a constant number 6.67 * 10-11 m3 kg-1 s-3
The universal gravitational constant.
What’s my Fgravity with Earth?
M1 = Me… 80 kgM2 = Earth… 5.97 * 1024 kgR = Radius of Earth… 6371 kmR = Radius of Earth… 6.371*106 m
…plug in the numbers…
784 Newtons
PYTS/ASTR 206 – Orbits and Gravity 10
Reality check
Which is greater, the attraction between you and the Earth or you and the Sun?
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Reality check
Which is greater, the attraction between you and the Earth or you and the Sun? The Earth – much closer…
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Reality check
Which is greater, the force on you from the Earth or force on the Earth from you?
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Reality check
Which is greater, the force on you from the Earth or force on the Earth from you? They’re the same – equal and opposite.
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It turns out that gravitational acceleration doesn’t depend on how big your mass is
More mass makes it harder to accelerate
…but… more mass also means a greater gravitational
attraction
Just like Galileo’s falling object experiment
Fgravity = M1 * agravity
Fgravity = G * M1 * MEARTH
R2 M1 * agravity = G * M1 * MEARTH
R2
agravity = G * MEARTH = 9.8 m/s2
R2
You can do this for any planet
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Gravity has a strong influence on big bodies Planets are round Particles that formed them we’re reorganized into a spherical shape
…and a weak influence on the small ones. Asteroids tend to be irregular Particles that formed them remain stuck together in haphazard shape
The shape of planetsThe shape of planets
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So gravity makes planets round… That’s not the end of the story Planets are also spinning – creates a centrifugal force All planets bulge at the equator
The faster the spin the bigger the bulge
What Jupiter would look like ifWe make it spin fasterWhat Jupiter currently looks like
EquatorSpinsFast
Polar areaSpinsSlow
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Combination of gravity and centrifugal forces
Planets mostly round Slightly flattened … but that’s still not the whole story
Planet Equatorial Radius
PolarRadius
Earth 6378 km 6357 km
Mars 3396 km 3376 km
Jupiter 71,492 km 66,854 km
Jupiter rotates in only 10 hours Very flattened
Difference in diameters is about the size of the Earth!
We can figure out the expected shape by assuming a fluid planet
The ‘Hydrostatic approximation’
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Also variation underground Mass excesses
Increases gravity
Mass deficits Decreases gravity
Earth looks a lot like this But not quite….
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What does flat mean? Even a fluid planet (flat) has a curved surface Flat means that the gravitational potential is the same
B & C here have the same potential, A has a higher potential
Earth has lumps and bumps (mass deficits and excesses) These lower and raise the equipotential surface A “flat surface” on the Earth is pretty bumpy – called the Geoid
A
B CLevel ground
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Geoid Varies by about ±100m from the idealized ellipsoid The view below is highly exaggerated You can walk in and out of a ‘hollow’ in the geoid and it will have appeared
flat The geoid is the definition of what is flat
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We saw already that gravity depends on distance
When R is small (close) Fgravity is big This is what causes tides
Tides on Earth are caused by the Sun and the Moon
TidesTides
Fgravity = G * M1 * MEARTH
R2
Close.Strong attraction.
Not so close.Weaker attraction.
Not close at all.Weakest
attraction.
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Easier to think of this from the point of view of the center of the Earth Which experiences the average amount of attraction
Earth & Moon rotate around common center of gravity
Each point experiences centrifugal and gravitational forces
Forces at balanced at the center of the Earth
(but not at the surface)
The tidal effect comes from the difference in these forces
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The tidal effect stretches out the planet Not the same as flattening – stretch is in one direction only Deforms the solid rocks Pulls the liquid oceans even more
Tidal bulge points towards the Moon The Sun causes a smaller tidal bulge
View from the top
View from the side
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High tide when you’re pointing toward the Moon (or Sun)
Rotation of the Earth means the ‘high’ and ‘low’ tides come and go
Solar and lunar tides can… Reinforce each other… Spring tide Almost cancel each other… Neap tide
(As usual) that’s not the whole story…
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Tidal bulge doesn’t move exactly with the Moon Earth is a big object Rocks are stiff & hard to deform
Earth rotates and carries the tidal bulge forward The tidal bulge can’t re-adjust fast enough to stay beneath the Moon
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The Earth’s bulge and the Moon can attract each other
Earth’s rotation slows down ~0.002 seconds/century
Moon speeds up And spirals away from the Earth ~1cm / year
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Mars & Phobos Opposite situation – Phobos orbits
faster than Mars spins Phobos is ahead of its tidal bulge The tidal bulge can’t re-adjust fast
enough to stay beneath the Phobos
MarsPhobos
Mars rotation speeds up
Phobos slows down And spirals towards Mars Its days are numbered
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Observations by Tycho Brahe Done without a telescope!
Kepler’s 3 laws Planetary orbits are ellipses Planets move faster when
closer to the sun The period of a planet’s orbit
is related to its size Big orbits take longer to complete
Orbits Orbits
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Law 1: Orbits are ellipses An ellipse has two foci The further apart the foci are the more
elongated the ellipse With a circle the foci are both in the center and
it’s not elongated at all
The sun is at one of these foci Means that planets are closer to the sun
at some times that others
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We describe how elongated the orbit is by the eccentricity (e). Earth eccentricity is 0.017 – practically a circle!
We describe the size of the orbit with the semi-major axis (a). Half the long-axis of the ellipse
a
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Law 2: Objects move fastest when at closest to the sun Perihelion distance = a*(1-e) Aphelion distance = a*(1+e)
PerihelionDistance
AphelionDistance
These parts of the orbittake the same amount of time
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Law 3: The relation between semi-major axis and period Back in Kepler’s day the periods of planets were easy to measure This law allowed them to be converted into semi-major axis
p is the period (usually seconds) a is the semi-major axis (usually meters) ksun depends on the mass of the Sun.
…but we can make this easier Use years for p Use AU for a Then ksun is just one - i.e. you can ignore it
Careful, this works for things orbiting the Sun
Not other objects
p2 = ksun * a3
p2 = a3
In years In AU
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How about an example? An asteroid takes 2 years to orbit the Sun. What is its semimajor axis?
We know
So:
Use your calculator to figure this out: 4 = a3 So: 41/3 = a The semi-major axis, a, turns out to be 1.59 AU
And another? Io orbits Jupiter every 1.77 days at a distance of 5.9 Jupiter Radii For reasons we’ll talk about later in the course Europa takes twice as long. What’s the size of Europa’s orbit in Jupiter radii?
We know that, for things orbiting Jupiter:
Putting in the Io values of 1.77 for ‘p’ and 5.9 for ‘a’, gives kJupiter = 0.01525 Using that with Europa’s period of 3.54 days: 3.542 = 0.01525 * a3 Then ‘a’ for Europa comes out to 9.4 Jupiter Radii
p2 = a3
22 = a3
p2 = kJupiter * a3
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And another example… Io orbits Jupiter every 1.77 days at a distance of 5.9 Jupiter Radii For reasons we’ll talk about later in the course Europa takes twice as
long. What’s the size of Europa’s orbit in Jupiter radii?
We know that, for things orbiting Jupiter:
Putting in the Io values of 1.77 for ‘p’ and 5.9 for ‘a’, gives kJupiter = 0.01525
Using that with Europa’s period of 3.54 days: 3.542 = 0.01525 * a3
Then ‘a’ for Europa comes out to 9.4 Jupiter Radii
Here we used units of Jupiter radii and days for length and time. Kjupiter depends on the mass of Jupiter, not the Sun
p2 = kJupiter * a3
PYTS/ASTR 206 – Orbits and Gravity 35
Gravity Newton and Galileo
Planetary Shape Flattening The Geoid
Tides Fate of the Moon
Orbits of planetary objects Kepler’s laws
In this lecture…In this lecture…
Next: Light and Heat from Planets and StarsNext: Light and Heat from Planets and Stars
Reading
Chapter 4 to revise this lecture
Chapter 5 for next Tuesday
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